Let S(p) represent the collection of meromorphic univalent functions f in the unit disc D which possess a simple pole at z=p(0<p<1) and meet the normalization f(0)=f′(0)-1=0. In this article, we determine bounds for Hermitian Toeplitz determinants whose entries are the Taylor coefficients of functions in S(p). Furthermore, we derive bounds for Hermitian Toeplitz determinants for two specific subclasses of S(p).

In this paper, we study the class A(p) which includes functions f that are meromorphic in the unit disk Δ and have a simple pole at z=p for some p∈(0,1) with the normalization f(0)=0=f′(0)-1. We establish a sufficient condition for functions in this class to be univalent. Making use of this condition, we introduce a subfamily of A(p) consisting of univalent functions satisfying a certain differential inequality in Δ. Next, we obtain a representation formula for such functions. Additionally, we establish necessary and sufficient conditions on the coefficients bn for functions f∈A(p) of the form (Formula presented.) to belong to this class. Furthermore, we determine sharp upper bounds for |bn| for all n≥2. Finally, we establish a sharp estimate for the Fekete-Szegö functional associated with the newly introduced subclass.

We obtain bounds for certain functionals defined on a class of meromorphic functions in the unit disc of the complex plane with a nonzero simple pole. These bounds are sharp in a certain sense. We also discuss possible applications of this result. Finally, we generalise the result to meromorphic functions with more than one simple pole.

Let Co(p) be the class of all functions f defined in the unit disc D having a simple pole at z = p where 0 < p < 1 and analytic in D {p} with f (0) = 0 = f ́(0) − 1 such that f maps D onto a domain whose complement with respect to the extended complex plane is a bounded convex set. These functions are called concave univalent functions. Each f ∈ Co(p) has the following Taylor expansion: (Farmula Presented) In this article, we first determine the regions of variability of the difference of successive coefficients (an+1 − an) for n ≽ 3 . We also find sharp upper bounds of the Toeplitz determinants, the entries of which are the Taylor coefficients of functions in Co(p) .

The distance matrix of a simple connected graph G is (Formula presented.) where dij is the length of a shortest path between the ith and jth vertices of G. Eigenvalues of D(G) are called the distance eigenvalues of G. The m-generation n-prism graph or (m, n)-prism graph can be defined in an iterative way where (Formula presented.) -prism graph is an n-vertex cycle. In this paper, we first find the number of zero eigenvalues of the distance matrix of a (m, n)-prism graph. Next, we find some quotient matrix that contains m nonzero distance eigenvalues of a (m, n)-prism graph. Our next result gives the rest of the nonzero distance eigenvalues of a (m, n)-prism graph in terms of distance eigenvalues of a cycle. Finally, we find the characteristic polynomial of the distance matrix of a (m, n)-prism graph. Applying this result, we provide the explicit distance eigenvalues of a (Formula presented.) -prism graph.

Prism networks belong to generalized petersen graph topologies with planar and polyhedral properties. These networks are extensively used to study the complex networks in the context of computer science, biological networks, brain networks, and social networks. In this letter, the performance analysis of consensus algorithms over prism networks is investigated in terms of convergence time, network coherence, and communication delay. Specifically, in this letter, we first derive the explicit expressions for eigenvalues of Laplacian matrix over m -dimensional prism networks using spectral graph theory. Subsequently, the study of consensus metrics such as convergence time, maximum communication time delay, first order network coherence, and second order network coherence is performed. Our results indicate that the effect of noise and communication time-delay on the consensus dynamics in m -dimensional prism networks is minimal. The obtained results illustrate that the scale-free topology of m -dimensional prism networks along with loopy structure is responsible for strong robustness with respect to consensus dynamics in prism networks.

Let Vp(λ) be the class of all functions f defined on the open unit disc D of the complex plane having simple pole at z= p, p∈ (0 , 1) and analytic in D { p} satisfying the normalizations f(0) = 0 = f′(0) - 1 such that | (z/ f(z)) 2f′(z) - 1 | < λ for z∈ D, λ∈ (0 , 1]. In this article, we obtain sharp bounds of the Zalcman and the generalized Zalcman functionals for functions in Vp(λ) for some indices of these functionals. As consequences of the obtained results, we pose the Zalcman-like coefficient conjectures for this class of functions. In addition, we estimate bound for the generalised Fekete–Szegö functional along with bounds of the second- and the third-order Hankel determinants for this class of functions.

Let V p (λ) be the collection of all functions f defined in the unit disc D having a simple pole at z= p where 0 < p< 1 and analytic in D {p} with f(0) = 0 = f ′ (0) - 1 and satisfying the differential inequality | (z/ f(z)) 2 f ′ (z) - 1 | < λ for z∈ D, 0 < λ≤ 1. Each f∈ V p (λ) has the following Taylor expansion: f(z)=z+∑n=2∞an(f)zn,|z|<p.We recently conjectured that |an(f)|≤1-(λp2)npn-1(1-λp2)forn≥3,while investigating functions in the class V p (λ). In the present article, we first obtain a representation formula for functions in this class. Using this, we prove the aforementioned conjecture for n= 3 , 4 , 5 whenever p belongs to certain subintervals of (0, 1). Also we determine non sharp bounds for |an(f)|,n≥3 and for |an+1(f)-an(f)/p|,n≥2.

Let Vp(λ) be the collection of all functions f defined in the open unit disk D, having a simple pole at z= p where 0 < p< 1 and analytic in D { p} with f(0) = 0 = f′(0) - 1 and satisfying the differential inequality | (z/ f(z)) 2f′(z) - 1 | < λ for z∈ D, 0 < λ≤ 1. Each f∈ Vp(λ) has the following Taylor expansion: f(z)=z+∑n=2∞anzn,|z|<p.In Bhowmik and Parveen (Bull Korean Math Soc 55(3):999–1006, 2018), we conjectured that |an|≤1-(λp2)npn-1(1-λp2)forn≥3,and the above inequality is sharp for the function kpλ(z)=-pz/(z-p)(1-λpz). In this article, we first prove the above conjecture for all n≥ 3 where p is lying in some subintervals of (0, 1). We then prove the above conjecture for the subordination class of Vp(λ) for p∈ (0 , 1 / 3]. Next, we consider the Laurent expansion of functions f∈ Vp(λ) valid in | z- p| < 1 - p and determine the exact region of variability of the residue of f at z= p and find the sharp bounds of the modulus of some initial Laurent coefficients for some range of values of p. The growth and distortion results for functions in Vp(λ) are also obtained. Next, we prove that Vp(λ) does not contain the class of concave univalent functions for λ∈ (0 , 1] and vice-versa for λ∈ ((1 - p2) / (1 + p2) , 1]. Finally, we show that there are some sets of values of p and λ for which C¯kpλ(D) may or may not be a convex set.

In this article we consider the class A(p) which consists of functions that are meromorphic in the unit disc D having a simple pole at z = p ∈ (0, 1) with the normalization f(0) = 0 = f′ (0)−1. First we prove some sufficient conditions for univalence of such functions in D. One of these conditions enable us to consider the class Vp(λ) that consists of functions satisfying certain differential inequality which forces univalence of such functions. Next we establish that (Formula presented), where Up(λ) was introduced and studied in [2]. Finally, we discuss some coefficient problems for Vp(λ) and end the article with a coefficient conjecture.

In this article, we consider a class denoted by A(P) which consists of functions f that are holomorphic in the unit disc ⅅ punctured at a point p ∈ (0, 1) where f has a simple pole. We prove a sufficient condition for these functions to be univalent in ⅅ. By using this condition, we construct the family Up(λ) of all functions f ∈ A(P)such that |(z/f (z))2f’ (z) − 1| < λμ where μ = ((1 − p)/(1 + p))2 for some 0 < λ ≤ 1, z ∈ ⅅ. Therefore, functions in the class Up(λ) are necessarily univalent. We present some basic properties for functions in the class Up(λ) which include an integral representation formula for such functions and obtain the exact region of variability of the second Taylor coefficient for functions in this class. We also obtain a sharp estimate for the Fekete–Szegö functional defined on the class Up(λ) along with a subordination result for functions in this family. In addition, we obtain some necessary and sufficient coefficient conditions involving the coefficients bn for functions f ∈ A(p) of the form (Formula presented.) to be in the class Up(λ). We have also obtained sharp bounds for |bn|, n ≥ 1. .

Let A(p) be the class consisting of functions f that are holomorphic in D{p}, p ∈ (0, 1) possessing a simple pole at the point z = p with nonzero residue and normalized by the condition f(0) = 0 = f′ (0) − 1. In this article, we first prove a sufficient condition for univalency for functions in A(p). Thereafter, we consider the class denoted by Σ(p) that consists of functions f ∈ A(p) that are univalent in D. We obtain the exact value for ∆(r, z/ f), where the Dirichlet integral ∆(r, z/ f) is given by max f∈Σ(p) ∫∫ ∆(r, z/ f) = |z|<r |(z/ f(z))′ |2 dx dy, (z = x+iy), 0 < r ≤ 1. We also obtain a sharp estimate for ∆(r, z/ f) whenever f belongs to certain subclasses of Σ(p). Furthermore, we obtain sharp estimates of the integral means for the aforementioned classes of functions.

In this article, we consider meromorphic univalent functions f in the unit disc of the complex plane having a simple pole at z = α ∈ (0, 1) with nonzero residue b at z = α. In 1969, P.N. Chichra proved an area theorem for such functions. In this note, we generalize this theorem and prove an interesting consequence of this result.